How do you simplify $sqrt205$ ?

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Answer 1 · 2018-05-19T19:27:36

$sqrt(205)$ is already in simplest form.

Note that:

$205 = 5 * 41$

has no square factors larger than $1$ . As a result, its square root cannot be simplified.

Using the algorithm described in https://socratic.org/s/aR4mTUXq we can find that the continued fraction expansion for $sqrt(205)$ is:

$sqrt(205) = [14;bar(3,6,1,4,1,6,3,28)]$

$color(white)(sqrt(205)) = 14+1/(3+1/(6+1/(1+1/(4+1/(1+1/(6+1/(3+1/(28+...))))))))$

From this we can find the smallest solution of Pell's equation for $205$ :

$39689^2 = 205 * 2772^2 + 1$

and related efficient rational approximation:

$sqrt(205) ~~ 39689/2772 ~~ 14.31782107$

How do you simplify sqrt205sqrt205?